Chapter 7.3, Problem 4P

Each of Problems 1 through 6 can be interpreted as describeing the interaction of two species with populations $x$ and $y$. in each of these problems, carry out the following steps

a) Draw a direction field and describe how solutions seem to behave.

b) Find the critical points.

c) For each critical points, find the corresponding linear sytem. Find the eigenvectors of the linear system, classify each critical points as to type, and determine whether it is asymototically stable, stable, or unstable.

d) Sketch thetrajectories in the neighbourhood of each critical points.

e) Compute and plot enough trajectories of the given system to show clearly the behaviour of the solutions.

f) Determine the limiting behaviour of $x$ and $y$ as $t\to \infty$, and interpret the result in terms of the populations of the two species.

$dx/dt=x(1.5-0.5x-y),$ $dx/dt=y(0.75-y-0.125x)$